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Numerology about the Apery Constant Zeta(3)

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  • WarrenS
    ... Hi. To follow up on SMR s post and do some crude numerology, I computed Zeta(3)/Pi^3 to 1000 decimals:
    Message 1 of 9 , Mar 14, 2011
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      --- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz <s_m_ruiz@...> wrote:
      >
      > I have obtained a curious
      > aproximation to Apery constant Zeta(3)=
      > 1.2020569031595942...
      >
      > (199/155)(16/165)^(3/2)Pi^3

      Hi.
      To follow up on SMR's post and do some crude numerology, I computed Zeta(3)/Pi^3 to 1000 decimals:
      0.038768179602916798941119890318721149806234568039552579223126762123777137012286855271851...
      and then computed its regular continued fraction expansion

      [0; 25, 1, 3, 1, 6, 3, 1, 1, 2, 1, 10, 3, 2, 1, 19, 3, 2, 1, 1, 3, 2, 3, 5, 3, 1, 1, 7, 1, 1, 1, 2, 1, 364, 11, 1, 84, 9, 34, 1, 7, 1, 63, 7, 1, 1, 4, 1, 5, 4, 7, 1, 1, 1, 5, 1, 4, 1, 5, 5, 9, 1, 21, 1, 9, 1, 1, 3, 2, 7, 1, 8, 5, 1, 7, 5, 2, 3, 1, 1, 1, 1, 1, 1, 1, 7, 8, 2, 2, 2, 1, 2, 1, 2, 6, 33, 99, 1, 1, 14, 1, 7, 2, 1, 1, 3, 4, 7, 1, 6, 5, 3, 1, 7, 8, 3, 5, 2, 1, 104, 1, 1, 1, 3, 8, 5, 1, 2, 1, 1, 11, 7, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 8, 1, 1, 7, 2, 4, 1, 1, 11, 1, 1, 1, 1, 35, 7, 1, 2, 3, 6, 1, 3, 1, 5, 81, 1, 2, 2, 7, 94, 2, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 3, 1, 10, 75, 1, 2, 6, 3, 2, 7, 1, 1, 1, 6, 1, 4, 1, 1, 10, 1, 8, 2, 2, 1, 2, 1, 5, 6, 2, 1, 3, 1, 5, 1, 4, 3, 2, 5, 3, 3, 1, 10, 3, 1, 1, 9, 1, 1, 3, 3, 4, 1, 1, 2, 2, 1, 1, 15, 1, 15, 7, 4, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 8, 1, 1, 2, 1, 2, 4, 3, 1, 1, 3, 12, 2, 3, 1, 10, 2, 5, 19, 1, 4, 4, 3, 1, 3, 1, 6, 1, 7, 1, 4, 1, 1, 2, 7, 94, 7, 4, 2, 1, 4, 1, 1, 2, 1, 2, 11, 2, 2, 1, 24, 2, 2, 1, 1, 6, 1, 1, 5, 2, 2, 1, 1, 1, 10, 1, 19, 1, 3, 1, 1, 1, 6, 1, 161, 1, 7, 1, 4, 5, 1, 5, 1, 2, 1, 4, 12, 1, 6, 2, 1, 7, 1, 4, 108, 1, 5, 1, 1, 3, 4, 4, 1, 1, 7, 1, 32, 1, 1, 1, 1, 2, 2, 3, 1, 1, 4, 2, 1, 4, 14, 1, 1, 2, 3, 16, 1, 3, 1, 2, 1, 1, 5, 1, 15, 2, 1, 2, 30, 1, 94, 7, 1, 1, 1, 1, 1, 1, 2, 2, 6, 6, 4, 2, 1, 4, 1, 5, 19, 1, 1, 1, 4, 8, 22, 1, 2, 3, 1, 5, 1, 1, 5, 2, 10, 6, 3, 4, 13, 1, 1, 1, 2, 1, 1, 97, 1, 1, 5, 2, 8, 1, 1, 2, 1, 3, 36, 1, 1, 2, 2, 1, 40, 1, 13, 3, 1, 1, 226, 1, 5, 3, 1, 7, 3, 19, 1, 20, 49, 1, 1, 33, 1, 7, 2, 1, 4, 7, 3, 7, 8, 1, 1, 55, 1, 17, 6, 1, 1, 1, 2, 1, 1, 5, 2, 8, 26, 3, 5, 6, 1, 2, 2, 2, 3, 1, 2, 2, 1, 33, 1, 3, 2, 2, 42, 1, 1, 1, 3, 2, 2, 1, 26, 1, 4, 7, 13, 7, 1, 29, 11, 1, 2, 1, 4, 1, 14, 3, 5, 2, 59, 6, 2, 2, 1, 103, 1, 8, 1, 3, 1, 2, 1, 3, 1, 3, 1, 1, 12, 3, 13, 2, 2, 1, 13, 1, 1, 40, 4, 1, 2, 3, 1, 3, 1, 1, 12, 1, 7, 1, 2, 1, 1, 49, 1, 2, 1, 3, 1, 3, 8, 1, 4, 1, 2, 5, 1, 3, 1, 7, 2, 2, 24, 3, 2, 3, 2, 11, 2, 1, 15, 1, 3, 54, 1, 1, 1, 1, 1, 8, 1, 5, 1, 2, 5, 6, 2, 7, 2, 1, 1, 8, 1, 1, 5, 1, 2, 2, 4, 3, 8, 2, 1, 1, 8, 43, 1, 1, 1, 17, 1, 3, 4, 24, 1, 6, 2, 490, 6, 3, 1, 3, 8, 1, 1, 27, 2, 2, 1, 2, 1, 18, 5, 5, 1, 2, 2, 1, 2, 1, 6, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 14, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 4, 1, 3, 1, 4, 1, 6, 3, 1, 1, 1, 2, 2, 26, 1, 1, 1, 1, 1, 1, 29, 2, 2, 2, 21, 6, 1, 2, 4, 70, 1, 50, 2, 1, 2, 1, 1, 3, 1, 3, 4, 1, 3, 3, 4, 2, 48, 1, 2, 1, 16, 1, 2, 2, 1, 32, 329, 3, 4, 12, 1, 3, 10, 2, 2, 1, 1, 1, 1, 1, 3, 3, 7, 400, 1, 3, 7, 4, 2, 9, 2, 2, 41, 1, 2, 75, 1, 12, 4, 2, 47, 32, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 2, 1, 23, 1, 112, 12, 1, 2, 1, 11, 5, 5, 3, 7, 1, 1, 1, 8, 1, 12, 1, 8, 1, 5, 4, 9, 1, 3, 1, 1, 3, 1, 5, 8, 1, 1, 4, 1, 2, 2, 1, 2, 38, 1, 2, 1, 2, 2, 38, 59, 1, 8, 4, 2, 1, 4, 2, 3, 1, 29, 19, 2, 369, 1, 1, 7, 1, 5, 9, 2, 3, 1, 1, 3, 2, 2, 34, 1, 1, 1, 11, 1, 3, 1, 8, 1, 10, 1, 1, 1, 32, 1, 3, 1, 1, 8, 4, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 8, 2, 3, 1, 15, 1, 2, 2, 1, 10, 5, 1, 2, 2, 4, 29, 8, 5, 1, 1, 2, 5, 1, 2, 1, 2, 6, 1, 1, 1, 11, 2, 5, 1, 2, 115, 6, 14, 4, 121, 1, 4, 2, 15, 34, 24, 6, 1, ...]

      Now we ask, based on this, "is Sebastien Martin Ruiz on to something interesting?"

      The regular continued fraction (RCF)
      expansion of a number X is useful for detecting
      if that number is rational -- which is equivalent to the RCF
      terminating, for example 355/113=[3;7,16].

      It also is useful for detecting if X is a quadratic irrational,
      which is equivalent to the RCF being eventually periodic, for
      example
      sqrt2=[1;2,2,2,2,...]
      and
      15^(3/2) = [58; repeat(10, 1, 1, 4, 8, 12, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 12, 8, 4, 1, 1, 10, 116)].

      Finally, it is useful for detecting if X is "unusual."
      Almost all real numbers have random RCF partial quotients
      sampled from the Gauss-Kuzmin distribution
      http://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution
      Any number which violently fails statistical tests for that
      is "unusual." For example
      e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14,...]

      Here are the counts A of how many times N appeared as a partial quotient for N=1,2,...,100:

      A=count for Zeta(3)/Pi^3 = 0.038768
      B=corresponding count for Pi+2^(1/3)+7^(1/2) = 7.047265...
      G=Gauss-Kuzmin expected count

      N, A, B, G
      1, 419, 402, 415.0374993
      2, 174, 185, 169.9250014
      3, 92, 94, 93.10940439
      4, 50, 60, 58.89368905
      5, 45, 36, 40.64198453
      6, 27, 28, 29.74734345
      7, 36, 21, 22.72007650
      8, 26, 14, 17.92190798
      9, 7, 14, 14.49956969
      10, 10, 15, 11.97264165
      11, 9, 10, 10.05366460
      12, 9, 11, 8.562013557
      13, 5, 8, 7.379530341
      14, 5, 6, 6.426269095
      15, 5, 9, 5.646563141
      16, 2, 5, 5.000681040
      17, 2, 6, 4.459648259
      18, 1, 4, 4.001930553
      19, 6, 7, 3.611253552
      20, 1, 4, 3.275132038
      21, 2, 5, 2.983858436
      22, 1, 2, 2.729792802
      23, 1, 1, 2.506855594
      24, 3, 4, 2.310160687
      25, 1, 1, 2.135744286
      26, 3, 2, 1.980364109
      27, 1, 1, 1.841346819
      28, 0, 2, 1.716472527
      29, 4, 3, 1.603885687
      30, 1, 0, 1.502025127
      31, 0, 0, 1.409570255
      32, 4, 3, 1.325397401
      33, 3, 2, 1.248546193
      34, 2, 1, 1.178191153
      35, 1, 0, 1.113620255
      36, 1, 0, 1.054216386
      37, 0, 0, 0.9994424311
      38, 2, 1, 0.9488293993
      39, 0, 1, 0.9019662943
      40, 2, 1, 0.8584915734
      41, 1, 2, 0.8180862026
      42, 1, 2, 0.7804680132
      43, 1, 1, 0.7453862058
      44, 0, 0, 0.7126180220
      45, 0, 0, 0.6819641182
      46, 0, 1, 0.6532465388
      47, 1, 0, 0.6263056815
      48, 1, 0, 0.6009976942
      49, 2, 1, 0.5771934627
      50, 1, 0, 0.5547760091
      51, 0, 0, 0.5336397688
      52, 0, 1, 0.5136888564
      53, 0, 0, 0.4948361984
      54, 1, 0, 0.4770028103
      55, 1, 0, 0.4601164965
      56, 0, 1, 0.4441111278
      57, 0, 0, 0.4289267856
      58, 0, 0, 0.4145080281
      59, 2, 0, 0.4008043244
      60, 0, 0, 0.3877691866
      61, 0, 0, 0.3753597376
      62, 0, 0, 0.3635365655
      63, 1, 0, 0.3522634356
      64, 0, 1, 0.3415067120
      65, 0, 0, 0.3312352143
      66, 0, 0, 0.3214203598
      67, 0, 0, 0.3120352996
      68, 0, 0, 0.3030553501
      69, 0, 1, 0.2944575600
      70, 1, 0, 0.2862205664
      71, 0, 0, 0.2783244500
      72, 0, 0, 0.2707507354
      73, 0, 1, 0.2634818134
      74, 0, 0, 0.2565019512
      75, 2, 1, 0.2497957052
      76, 0, 1, 0.2433490748
      77, 0, 1, 0.2371487818
      78, 0, 0, 0.2311825580
      79, 0, 0, 0.2254387130
      80, 0, 0, 0.2199062779
      81, 1, 0, 0.2145750055
      82, 0, 0, 0.2094353705
      83, 0, 0, 0.2044781363
      84, 1, 0, 0.1996947880
      85, 0, 0, 0.1950773880
      86, 0, 1, 0.1906182871
      87, 0, 0, 0.1863104142
      88, 0, 0, 0.1821469860
      89, 0, 0, 0.1781215085
      90, 0, 0, 0.1742280648
      91, 1, 0, 0.1704607381
      92, 0, 0, 0.1668144771
      93, 0, 0, 0.1632839428
      94, 3, 0, 0.1598642283
      95, 0, 0, 0.1565508601
      96, 0, 0, 0.1533395088
      97, 1, 1, 0.1502259897
      98, 0, 0, 0.1472062620
      99, 1, 0, 0.1442767180
      100, 0, 0, 0.1414337502

      "Chi Squared" values for departure of observed counts from Gauss-Kuzmin law:

      for A: 560.1
      for B: 536.4

      I played around a bit more and found ChiSquared for several more "random numbers" like B, for example
      Pi-2^(1/3)+7^(1/2)-exp(1)-ln(3)+erf(1) = 1.55322959...
      has ChiSquared=771.9.
      So Ruiz's number in (A) is within the typical range of ChiSquared.

      In contrast, e has enormous ChiSquared=107491
      or sqrt(7) has ChiSquared=189729, both easily detected in this
      way as "unusual."

      CONCLUSION:
      Sorry, the first 1000 RCF quotients do not seem to indicate that
      Zeta(3)/Pi^3 is in any way an unusual real number.

      You all might want to try larger statistical analyses than this,
      but that's what I'm seeing from the first 1000. (It might be good to create a fast "unusual number detector" software tool based on
      the above techniques and maybe a Kolmogorov-Smirnov test.)
    • WarrenS
      I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3 was a low-degree low-height algebraic number. Result: If it has degree
      Message 2 of 9 , Mar 14, 2011
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        I also attempted to use PSLQ to figure out whether
        Zeta(3)/Pi^3
        was a low-degree low-height algebraic number.
        Result:
        If it has degree<=10 then its height is at least 10^91.

        So, sorry again:
        Zeta(3)/Pi^3 again fails to look unusual.
      • Jack Brennen
        No magic here, just the concept that you can find approximations to basically anything if you have enough degrees of freedom. For instance:
        Message 3 of 9 , Mar 14, 2011
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          No magic here, just the concept that you can find approximations to
          basically anything if you have enough degrees of freedom.

          For instance:

          (83/779)(605/689)^(3/2)Pi^3 == 2.71828182845...

          gives you the first 12 correct digits of e.

          (121/634)(305/724)^(3/2)Pi^3 == 1.6180339887...

          gives you the first 11 correct digits of the golden ratio (1+sqrt(5))/2.


          Note that there's an even better approximation to Zeta(3) at:

          (25/186)(86/197)^(3/2)Pi^3


          On 3/14/2011 9:59 AM, WarrenS wrote:
          >
          >
          > --- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz<s_m_ruiz@...> wrote:
          >>
          >> I have obtained a curious
          >> aproximation to Apery constant Zeta(3)=
          >> 1.2020569031595942...
          >>
          >> (199/155)(16/165)^(3/2)Pi^3
          >
          > Hi.
          > To follow up on SMR's post and do some crude numerology, I computed Zeta(3)/Pi^3 to 1000 decimals:
          > 0.038768179602916798941119890318721149806234568039552579223126762123777137012286855271851...
          > and then computed its regular continued fraction expansion
          >
          > [0; 25, 1, 3, 1, 6, 3, 1, 1, 2, 1, 10, 3, 2, 1, 19, 3, 2, 1, 1, 3, 2, 3, 5, 3, 1, 1, 7, 1, 1, 1, 2, 1, 364, 11, 1, 84, 9, 34, 1, 7, 1, 63, 7, 1, 1, 4, 1, 5, 4, 7, 1, 1, 1, 5, 1, 4, 1, 5, 5, 9, 1, 21, 1, 9, 1, 1, 3, 2, 7, 1, 8, 5, 1, 7, 5, 2, 3, 1, 1, 1, 1, 1, 1, 1, 7, 8, 2, 2, 2, 1, 2, 1, 2, 6, 33, 99, 1, 1, 14, 1, 7, 2, 1, 1, 3, 4, 7, 1, 6, 5, 3, 1, 7, 8, 3, 5, 2, 1, 104, 1, 1, 1, 3, 8, 5, 1, 2, 1, 1, 11, 7, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 8, 1, 1, 7, 2, 4, 1, 1, 11, 1, 1, 1, 1, 35, 7, 1, 2, 3, 6, 1, 3, 1, 5, 81, 1, 2, 2, 7, 94, 2, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 3, 1, 10, 75, 1, 2, 6, 3, 2, 7, 1, 1, 1, 6, 1, 4, 1, 1, 10, 1, 8, 2, 2, 1, 2, 1, 5, 6, 2, 1, 3, 1, 5, 1, 4, 3, 2, 5, 3, 3, 1, 10, 3, 1, 1, 9, 1, 1, 3, 3, 4, 1, 1, 2, 2, 1, 1, 15, 1, 15, 7, 4, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 8, 1, 1, 2, 1, 2, 4, 3, 1, 1, 3, 12, 2, 3, 1, 10, 2, 5, 19, 1, 4, 4, 3, 1, 3, 1, 6, 1, 7, 1, 4, 1, 1, 2, 7, 94, 7, 4, 2, 1, 4, 1, 1, 2, 1, 2, 11, 2, 2, 1, 24, 2, 2, 1, 1, 6, 1, 1, 5, 2, 2, 1,
          1, 1, 10, 1, 19, 1, 3, 1, 1, 1, 6, 1, 161, 1, 7, 1, 4, 5, 1, 5, 1, 2, 1, 4, 12, 1, 6, 2, 1, 7, 1, 4, 108, 1, 5, 1, 1, 3, 4, 4, 1, 1, 7, 1, 32, 1, 1, 1, 1, 2, 2, 3, 1, 1, 4, 2, 1, 4, 14, 1, 1, 2, 3, 16, 1, 3, 1, 2, 1, 1, 5, 1, 15, 2, 1, 2, 30, 1, 94, 7, 1, 1, 1, 1, 1, 1, 2, 2, 6, 6, 4, 2, 1, 4, 1, 5, 19, 1, 1, 1, 4, 8, 22, 1, 2, 3, 1, 5, 1, 1, 5, 2, 10, 6, 3, 4, 13, 1, 1, 1, 2, 1, 1, 97, 1, 1, 5, 2, 8, 1, 1, 2, 1, 3, 36, 1, 1, 2, 2, 1, 40, 1, 13, 3, 1, 1, 226, 1, 5, 3, 1, 7, 3, 19, 1, 20, 49, 1, 1, 33, 1, 7, 2, 1, 4, 7, 3, 7, 8, 1, 1, 55, 1, 17, 6, 1, 1, 1, 2, 1, 1, 5, 2, 8, 26, 3, 5, 6, 1, 2, 2, 2, 3, 1, 2, 2, 1, 33, 1, 3, 2, 2, 42, 1, 1, 1, 3, 2, 2, 1, 26, 1, 4, 7, 13, 7, 1, 29, 11, 1, 2, 1, 4, 1, 14, 3, 5, 2, 59, 6, 2, 2, 1, 103, 1, 8, 1, 3, 1, 2, 1, 3, 1, 3, 1, 1, 12, 3, 13, 2, 2, 1, 13, 1, 1, 40, 4, 1, 2, 3, 1, 3, 1, 1, 12, 1, 7, 1, 2, 1, 1, 49, 1, 2, 1, 3, 1, 3, 8, 1, 4, 1, 2, 5, 1, 3, 1, 7, 2, 2, 24, 3, 2, 3, 2, 11, 2, 1, 15, 1, 3, 54, 1, 1, 1, 1, 1, 8, 1, 5, 1, 2, 5, 6
          , 2, 7, 2, 1, 1, 8, 1, 1, 5, 1, 2, 2, 4, 3, 8, 2, 1, 1, 8, 43, 1, 1, 1
          > , 17, 1, 3, 4, 24, 1, 6, 2, 490, 6, 3, 1, 3, 8, 1, 1, 27, 2, 2, 1, 2, 1, 18, 5, 5, 1, 2, 2, 1, 2, 1, 6, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 14, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 4, 1, 3, 1, 4, 1, 6, 3, 1, 1, 1, 2, 2, 26, 1, 1, 1, 1, 1, 1, 29, 2, 2, 2, 21, 6, 1, 2, 4, 70, 1, 50, 2, 1, 2, 1, 1, 3, 1, 3, 4, 1, 3, 3, 4, 2, 48, 1, 2, 1, 16, 1, 2, 2, 1, 32, 329, 3, 4, 12, 1, 3, 10, 2, 2, 1, 1, 1, 1, 1, 3, 3, 7, 400, 1, 3, 7, 4, 2, 9, 2, 2, 41, 1, 2, 75, 1, 12, 4, 2, 47, 32, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 2, 1, 23, 1, 112, 12, 1, 2, 1, 11, 5, 5, 3, 7, 1, 1, 1, 8, 1, 12, 1, 8, 1, 5, 4, 9, 1, 3, 1, 1, 3, 1, 5, 8, 1, 1, 4, 1, 2, 2, 1, 2, 38, 1, 2, 1, 2, 2, 38, 59, 1, 8, 4, 2, 1, 4, 2, 3, 1, 29, 19, 2, 369, 1, 1, 7, 1, 5, 9, 2, 3, 1, 1, 3, 2, 2, 34, 1, 1, 1, 11, 1, 3, 1, 8, 1, 10, 1, 1, 1, 32, 1, 3, 1, 1, 8, 4, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 8, 2, 3, 1, 15, 1, 2, 2, 1, 10, 5, 1, 2, 2, 4, 2
          9, 8, 5, 1, 1, 2, 5, 1, 2, 1, 2, 6, 1, 1, 1, 11, 2, 5, 1, 2, 115, 6, 14, 4, 121, 1, 4, 2, 15, 34, 24, 6, 1, ...]
          >
        • djbroadhurst
          ... We should always try to do what most folk think can never be done. Had such an attempt succeeded, there would be several deeply unhappy Fields Medalists,
          Message 4 of 9 , Mar 14, 2011
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            --- In primenumbers@yahoogroups.com,
            "WarrenS" <warren.wds@...> wrote:

            > I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3
            > was a low-degree low-height algebraic number.

            We should always try to do what most folk think can never be done.

            Had such an attempt succeeded, there would be several deeply
            unhappy Fields Medalists, who believe (as I do) in the
            Drinfeld-Deligne conjecture that the Grottendieck-Teichmueller
            algebra has precisely one generator in each odd degree.

            Yet that is, emphatically, not a reason for avoiding serious
            experiments to investigate a contrary hypothesis. To the best of
            my recollection, the possibility that zeta(3)/Pi^3 might be an
            algebraic number has been investigated to many tens of thousands
            of decimal digits of numerical precision. However, I am unable
            to provide a reference. Might someone else do so, please?

            David
          • WarrenS
            ... --well, you (DJB) probably have better PSLQ and better hardware than I do (I bet) and could do so yourself... I see Broadhurst wrote a quantum field theory
            Message 5 of 9 , Mar 15, 2011
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              > > Warren D Smith:
              > > I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3
              > > was a low-degree low-height algebraic number.

              > DJ Broadhurst:
              > Had such an attempt succeeded, there would be several deeply
              > unhappy Fields Medalists, who believe (as I do) in the
              > Drinfeld-Deligne conjecture that the Grottendieck-Teichmueller
              > algebra has precisely one generator in each odd degree.
              >
              > Yet that is, emphatically, not a reason for avoiding serious
              > experiments to investigate a contrary hypothesis. To the best of
              > my recollection, the possibility that zeta(3)/Pi^3 might be an
              > algebraic number has been investigated to many tens of thousands
              > of decimal digits of numerical precision. However, I am unable
              > to provide a reference.

              --well, you (DJB) probably have better PSLQ and better hardware
              than I do (I bet) and could do so yourself...

              I see Broadhurst wrote a quantum field theory paper "where do the tedious products of zetas come from?" (actual title!) which mentions the "Drinfeld Deligne conjecture"
              and "Grothendieck¬ĖTeichm:uller algebra" (whatever they are) as well as using PSLQ.
              Unfortunately non-experts (i.e. me) will have a difficult time understanding this paper.
              Even after (trying to) read it, I still have almost no clue what DDC and GTA are,
              and it gives zero cites to D,D,G and T's work.

              Anyhow I agree Zeta(3)/Pi^3 being an "unusual number" was a pretty low-chance proposition, but SM Ruiz's post stimulated me to look, and sure enough, negative result.
              About the idea that you can produce miracles on demand if you have enough fitting freedom... well, yes, but the approach I used attempts to quantify the miraculousness,
              and the TRUE miracles yield far better approximations than the amount of fitting freedom would suggest. Don't accept lame miracles, demand real miracles.
            • djbroadhurst
              ... An editor solicited this article and I agreed on the strict condition that tedious be retained in the title. It happened that I needed to consult it
              Message 6 of 9 , Mar 15, 2011
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                --- In primenumbers@yahoogroups.com,
                "WarrenS" <warren.wds@...> wrote:

                > Broadhurst wrote a quantum field theory paper
                > "where do the tedious products of zetas come from?"

                An editor solicited this article and I agreed on the
                strict condition that "tedious" be retained in the title.
                It happened that I needed to consult it yesterday
                and fortunately it was easy to google:
                > Ungefähr 84 Ergebnisse (0,18 Sekunden)

                David
              • WarrenS
                ... --for example (this is not due to me; perhaps it was first noticed by Ramanujan?) exp(Pi * sqrt(n)) is within 10^(-12) of being an integer if n=163. This
                Message 7 of 9 , Mar 15, 2011
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                  > About the idea that you can produce miracles on demand if you have enough fitting freedom... well, yes, but the approach I used attempts to quantify the miraculousness,
                  > and the TRUE miracles yield far better approximations than the amount of fitting freedom would suggest. Don't accept lame miracles, demand real miracles.


                  --for example (this is not due to me; perhaps it was first noticed by Ramanujan?)
                  exp(Pi * sqrt(n))
                  is within 10^(-12) of being an integer if n=163.

                  This is beyond what any kind of fitting contrivance would be expected to produce.
                  Also, no other n<25000 gets you even within a 100,000 times further away
                  from being an integer.

                  PSLQ has in fact discovered truths valid not to 12, but to
                  an infinite number of decimals :)
                • mikeoakes2
                  ... Even better: (exp(Pi*sqrt(163))-744)^(1/3) = 640319.999999999999999999999999390... See Cohen, CCANT, p.383, which explains where astonishing results such
                  Message 8 of 9 , Mar 16, 2011
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                    --- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
                    >
                    > --for example (this is not due to me; perhaps it was first noticed by Ramanujan?)
                    > exp(Pi * sqrt(n))
                    > is within 10^(-12) of being an integer if n=163.
                    >
                    > This is beyond what any kind of fitting contrivance would be expected to produce.
                    > Also, no other n<25000 gets you even within a 100,000 times further away
                    > from being an integer.

                    Even better:
                    (exp(Pi*sqrt(163))-744)^(1/3) = 640319.999999999999999999999999390...

                    See Cohen, CCANT, p.383, which explains where astonishing results such as these come from.

                    Mike
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